Wave Propagation and Long-Time Behavior on the Driven Sine-Gordon Chain

Abstract

Model one-dimensional lattices have been an important tool in the study of nonlinear wave motion in solids^1,2,3. A particular model is specified by the choice of a potential function (usually restricted to nearest neighbors). The free end of a semi-infinite chain (the piston particle) is driven into the lattice at fixed speed to simulate some external impulse. The motion of the piston particle induces the creation of a propagating wave front. In these systems, the wavefront has the nature of a kink or soliton in that its speed depends on its amplitude and it can propagate (more or less) independently of radiation (phonons). A particularly careful study has been made of the Toda chain³, the only known completely integrable lattice system. The dynamical behavior observed in the Toda chain is apparently generic to all of these hard-core nearest-neighbor lattices. At low piston particle speeds the asymptotic behavior of the chain is harmonic-oscillator-like in the sense that the particles eventually come to rest in the piston particle frame of reference. At high piston particle speeds (or equivalently large anharmonicity) The asymptotic dynamics is hard-rod-like in the sense that the oscillations in the piston particle frame of reference persist.